[Completeness Assumption on Outcome-Inducing Proxy [1]}, [2]}]assumcompletenessproxy 
Let \(l: \mathcal {W} \rightarrow \mathbb {R}\)  be any square integrable function \(\Vert l\Vert _{{W}} < \infty \) , We assume the following condition:

l(W) A=a, Z=z=0   (a,z) A Z       l(w) = 0    W-a.e.


This assumption is not necessary for identification (Proposition REF ), but we need it for connecting the two-stage loss and the deviation of the bridge function. Given this assumption, we have the following consistency result.
